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Theorem distgp 24107
Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
distgp.1 𝐵 = (Base‘𝐺)
distgp.2 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
distgp ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)

Proof of Theorem distgp
StepHypRef Expression
1 simpl 482 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp)
2 simpr 484 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵)
3 distgp.1 . . . . . 6 𝐵 = (Base‘𝐺)
43fvexi 6920 . . . . 5 𝐵 ∈ V
5 distopon 23004 . . . . 5 (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵))
64, 5ax-mp 5 . . . 4 𝒫 𝐵 ∈ (TopOn‘𝐵)
72, 6eqeltrdi 2849 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵))
8 distgp.2 . . . 4 𝐽 = (TopOpen‘𝐺)
93, 8istps 22940 . . 3 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
107, 9sylibr 234 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp)
11 eqid 2737 . . . . . 6 (-g𝐺) = (-g𝐺)
123, 11grpsubf 19037 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1312adantr 480 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
144, 4xpex 7773 . . . . 5 (𝐵 × 𝐵) ∈ V
154, 14elmap 8911 . . . 4 ((-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)) ↔ (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1613, 15sylibr 234 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)))
172, 2oveq12d 7449 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵))
18 txdis 23640 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵))
194, 4, 18mp2an 692 . . . . . 6 (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)
2017, 19eqtrdi 2793 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵))
2120oveq1d 7446 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽))
22 cndis 23299 . . . . 5 (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2314, 7, 22sylancr 587 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2421, 23eqtrd 2777 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2516, 24eleqtrrd 2844 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
268, 11istgp2 24099 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
271, 10, 25, 26syl3anbrc 1344 1 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  𝒫 cpw 4600   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Basecbs 17247  TopOpenctopn 17466  Grpcgrp 18951  -gcsg 18953  TopOnctopon 22916  TopSpctps 22938   Cn ccn 23232   ×t ctx 23568  TopGrpctgp 24079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-topgen 17488  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cn 23235  df-cnp 23236  df-tx 23570  df-tmd 24080  df-tgp 24081
This theorem is referenced by: (None)
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